Master in Instrumentation and Informatics in Physics, Astronomy and Space Research

SIDEBAND RATIO IN DOUBLE

SIDEBAND AND SIDEBAND-

SEPARATING RECEIVERS WITH A

MICHELSON INTERFEROMETER

Michela Romanini

Supervisors: Dr. A. M. Baryshev Dr. R. Hesper Dr. F. P. Mena Prof. dr. W. Wild

Groningen, 25 July 2007

2

TABLE OF CONTENTS

ABSTRACT.................................................................................................................... 4

Chapter 1. INTRODUCTION 5

1.1 Scientific background......................................................................... 5

1.1.1 Ground-based observations................................................. 6

1.2 Detection requirements...................................................................... 8

1.2.1 Direct detection...................................................................... 8

1.2.2 Heterodyne detection............................................................ 9

1.3 Sideband ratio...................................................................................... 11

1.4 Thesis goal and outline....................................................................... 12

Chapter 2. SUPERCONDUCTOR-BASED MIXERS 13

2.1 Phenomenology of superconductivity............................................. 13

2.2 Superconductor-Insulator-Superconductor junction.................... 15

2.2.1 Photon-assisted tunnelling in a SIS junction...................... 18

2.2.2 SIS junction as mixing element............................................ 18

2.2.3 Double sideband (DSB) mixers........................................... 19

2.2.4 Sideband-separating (2SB) mixers....................................... 21

Chapter 3. INSTRUMENTAL SETUP 25

3.1 The Fourier Transform Spectrometer: general description......... 25

3.2 Instrumental setup for double sideband mixers............................. 28

3.2.1 Local oscillator........................................................................ 29

3.2.2 Lock-in amplifier.................................................................... 30

3.2.3 Design of the parabolic mirrors........................................... 30

3.3 Instrumental setup for sideband-separating mixers....................... 32

3

Chapter 4. SIDEBAND RATIO OF DOUBLE SIDEBAND

MIXERS

35

4.1 Detection with double sideband mixers.......................................... 35

4.1.1 Direct detection measurements............................................ 35

4.1.2 Heterodyne detection measurements: mixer 1.................. 37

4.1.3 Heterodyne detection measurements: mixer 2.................. 39

4.2 Sideband ratio calculation.................................................................. 41

4.2.1 SBR results.............................................................................. 41

4.2.2 Discussion of the results....................................................... 43

4.3 Conclusions.......................................................................................... 46

Chapter 5. SIDEBAND RATIO OF SIDEBAND-SEPARATING

MIXERS

47

5.1 Description of the standard method................................................ 47

5.2 Heterodyne detection measurements.............................................. 49

5.3 Image rejection calculations.............................................................. 50

5.4 Conclusions.......................................................................................... 52

Appendix A: SBR error calculation................................................................... 53

Appendix B: programs with Mathematica 5.2.................................................

55

References.............................................................................................................

61

Acknowledgements.................................................................................................... 63

4

ABSTRACT

Terahertz heterodyne receivers typically use double sideband (DSB) or sideband-

separating (2SB) mixers. The precise knowledge of the receiver sideband ratio (SBR)

is a fundamental requirement for the calibration of the data taken with this type of

receivers. At the moment the spectroscopic techniques developed for the

determination of the SBR of DSB mixers (such as Martin Pupplet interferometry

and Gas cell technique) and 2SB mixer (such as the technique described by Kerr)

are rather complicated. Here, we present sideband ratio measurements of a DSB

and a 2SB receiver in the 600-720 GHz band (ALMA Band 9) using a Michelson

interferometer as input filter. The main requirement for this method is that the

resolution must be high enough to allow distinguishing between the upper and

lower sidebands. The advantages of this method are, first, the simplicity of the

experimental setup, and, second, the possibility to identify and calibrate out standing

waves in the signal and local oscillator paths. For DSB receivers, our procedure

allows using exactly the same receiver configuration for both direct and heterodyne

detections.

Based on our results on the DSB receiver we estimate that the direct detection can

be used to predict the SBR if an accuracy of 20% is sufficient. Also for the 2SB

receiver we found a satisfactory agreement between the SBR calculated from the

direct detection and the SBR calculated with the method described by Kerr. Further

experiments are needed to improve and better understand the results for both types

of mixers.

5

Chapter 1

INTRODUCTION

1.1 Scientific background

The (sub)millimeter regions (30 GHz-1 THz) of the electromagnetic spectrum, still

largely unexplored for astrophysical purposes, are the most promising ones for radio

astronomy and for measurements of atmospheric molecules [1].

The cold material (10-30K) which characterizes the early stages of star and planet

formation and the earliest stages in galaxy formation has its peak emission in the

sub-millimeter region. The origin and evolution of galaxies are closely tied to the

process in which stars eject gas and dust into the interstellar medium, while at the

same time gas and dust clouds in the interstellar medium collapse gravitationally to

form stars and/or planets [2]. All these processes are accompanied by absorption

and emission of radiation at sub-mm wavelengths. From such absorption and

emission spectra, detailed information on the dynamics, structure, and elemental and

isotopic abundances of these objects can be extracted, as well as on their

fundamental physical properties such as density and temperature [3].

The spectral lines emitted by a single atom are associated with changes in energies

of its orbiting electrons. Spectral lines of molecules are related not only to electronic

energy levels, but also to molecular motions. The large majority of the rotational and

vibrational transitions of molecules have characteristic wavelengths shorter than 1

cm. The spectral lines observed in atomic and molecular absorption and emission

spectra correspond to transitions occurring between different energy levels, and

have an intensity which depends on the transition probability, on the population of

the states, and on the relative abundance of the given chemical species. Radiative

transitions between different states may occur due to spontaneous emission,

collisions between atoms and molecules, and to the interaction with electromagnetic

6

radiation and with charged particles found in interstellar space. The measurement of

the intensity of different lines of the same molecular species allows assessing the

physical parameters of the emitting gas, such as velocity (Doppler shift),

temperature and pressure (broadness of the line). Moreover, the comparison of lines

characteristic of different molecules provides information about the local chemical

composition of the gas (along the line of sight).

Contrary to that of molecules, the emission spectrum of interstellar dust is

continuous (black body radiation) and varies smoothly with frequency, with a higher

intensity towards shorter wavelengths (Planck’s law). Since the dust temperature is

in the 10-100 K range, the dust emission spectrum lies in the sub-millimeter region

[4].

The sub-mm range is useful also for other purposes. The study of the cosmic

background radiation at sub-mm wavelengths can be used to obtain information

about the Big Bang [2]. Also the chemistry of the terrestrial atmosphere can be

studied in the sub-mm range through the observation of rotational transitions of gas

molecules. This is useful, for example, to obtain information about the origin and

progress of the greenhouse effect and the ozone depletion [5]. Other applications of

sub-millimeter and Terahertz technology are medical, healthcare and security [1].

1.1.1 Ground-based observations

A problem to overcome for ground-based observations in the mm and sub-mm

range lies in the fact that especially the water vapour has many strong absorption

lines in the sub-mm band which, because of the pressure broadening, may

completely obscure sources beyond the atmosphere. Therefore very dry conditions

at high altitudes are necessary to be able to observe stellar sources from Earth.

Figure 1.1 shows the atmospheric transmission at the Chajnantor site (5000 m

altitude). It is seen that the sub-mm band can be divided into several sub-bands,

where the atmosphere is relatively transparent. Inside these sub-bands the

observations from Earth are possible. It should be noted that, even on the best

7

terrestrial sites, observation of water lines is still impossible. For that a spacecraft or

a high-flying aircraft or balloon is needed.

The so-called Atacama Large Millimeter Array (ALMA), located at the Chajnantor

site, is the largest array of telescopes to function as an interferometer. After

completion the ALMA array will consist of 50 to 66 high-precision movable radio

antennas, each 12 metres in diameter, placed in a variable configuration. The

receivers cover the frequency range between approximately 30 and 1000 GHz split

over 10 bands fitting the atmospheric windows (fig. 1.1), with an angular resolution

from 0.1 to 0.01 arcsec at the highest bands with the greatest extent of the array.

Figure 1.2 shows the comparison between the angular resolution of ALMA and that

of other existing or planned major astronomical facilities, such as the Very Large

Array (VLA), the Hubble telescope (HST) or the Very Large Telescope (VLT).

Figure 1.1: Atmospheric transmission at the Chajnantor site (5000 m altitude) together

with the planned ALMA frequency bands. In winter (June-September), the

precipitable water vapour column (PVW) is ≤ 0.5 mm for about 25% of the

time (after Ref. 6).

8

Figure 1.2: Angular resolution of ALMA compared to other major astronomical facilities

(after Ref. 7).

1.2 Detection requirements

To obtain information on the variety and abundance of gas components present in

the universe or in the Earth’s atmosphere, a full spectrum over a wide wavelength

range is necessary. Two basic method of detection can be used: direct detection and

heterodyne detection.

1.2.1 Direct detection

Direct detection refers to a process in which a photon either raises the temperature

of a bolometer element or causes an electron current to flow in a photoconductor.

This is a power detection process in which the fundamental noise limiting the

sensitivity is the signal noise. Due to limiting properties of the optics and the

detector itself, the frequency band over which the sub-millimeter radiation can be

observed is limited. Direct detectors are not able to resolve the phase of the

detected radiation. Moreover, a direct detector can not distinguish between two

signals of equal power and different radiation frequencies. Then, in order to do

spectroscopy, it is necessary put a band-pass filter in front of the detector, such as a

9

Michelson or a Martin-Pupplett interferometer. The problem with a typical

spectrometer is that the spectral resolution (λ/L) is limited by the size L of the

spectrometer. This is the reason why direct detection has limited applications in sub-

millimeter spectroscopy when high frequency resolution and sensitivity are required.

However, for broad-band mapping purposes they are excellent.

1.2.2 Heterodyne detection

To overcome the problems described above, heterodyne detection is used. In a

heterodyne measurement, a strong local oscillator (LO) signal, of frequency close to

the frequency to be detected, is combined with the weak astronomical signal from

the antenna and then directed onto a non-linear detector. This detector then mixes

the incoming waves and provides an intermediate frequency (IF) at the output of

few GHz. The components of the down-converted signal can be separated and

integrated to obtain the signal frequency spectrum.

To better explain the concept of the heterodyne mixing, let us consider a non-linear

element with a current-voltage dependence given by I(V)=V2. The incoming

radiation and local oscillator signals can be represented by sinusoidal voltages:

)sin()(

)sin()(

tVtV

tVtV

LOLOLO

SSS

ω

ω

=

= (1.1)

The resulting current is given by:

[ ]2

)sin()sin( tVtVI LOLOSS ωω += (1.2)

[ ] [ ] [ ]ttVVtV

tV

LOSLOSSLOLOLO

SS )cos()cos()2cos(1

2)2cos(1

2

22

ωωωωωω −+++−+−=

The signal is then passed through a band filter around the IF frequency, which is

defined as the difference of the LO and antenna frequencies.

10

Figure 1.3: Conversion of the signal (ωLO+ωIF) and image (ωLO−ωIF) band during the

heterodyne mixing process. The “negative” frequencies on the horizontal axis

(left of the origin) are equivalent to their absolute value (right of the origin). In

this example, the LO frequency is 1000 GHz, while the IF band is centred at

2 GHz.

Because negative frequencies are equal to positive ones, any down-conversion of the

antenna frequency ωS for which ±(ωS−ωLO) is within the IF band passes through the

filter. Hence, the IF output signal at a frequency ωIF can originate from a signal at

either of the two frequencies ωS=ωLO ± ωIF (fig. 1.3).

Since both the frequencies above and below the LO frequency are converted into

the same IF band, the detector is said to operate in double side band (DSB) mode.

The band at a frequency above the LO frequency is called upper sideband (USB)

and the band at a frequency below the LO frequency is called lower sideband (LSB).

Three bands of interest can be described:

• The radio frequency (RF) bandwidth, which contains the frequency of the

antenna signal. The detector is sensitive to this frequency range.

• The IF band around the IF frequency (the centre frequency to which the

signal is converted). The IF bandwidth determines how many spectral lines

are observed at one time. The IF bandwidth is set by a band-pass filter at the

IF output of the detector or limited by the detector itself.

• The LO bandwidth (or tuning range), which determines the frequencies that

can actually be observed.

11

Figure 1.4 shows a block diagram of a heterodyne receiving system. The antenna

signal is coupled with the LO signal and the spectral components of the combined

beam are down-converted in the mixer. The IF signal is amplified and then analyzed

by the back-end spectrometer or by a correlator. Heterodyne mixing requires a non-

linear element such as a Schottky-barrier diode, a Superconductor-Insulator-

Superconductor (SIS) junction or a Hot Electron Bolometer (HEB). ALMA Band 9

works with SIS junction as a mixer. Detailed information about this kind of junction

is presented in section 2.2.

Figure 1.4: block diagram of heterodyne receiving system.

1.3 Sideband ratio

The output current of the mixer at a frequency ωIF contains two contributions,

arising from the frequency components of the reference signal intensity at ωLO+ωIF

and ωLO–ωIF, as for both these frequencies the absolute frequency difference with

respect to the fixed LO frequency is within the IF band (see fig. 1.3).

Hence, the accurate calibration of a heterodyne receiver requires knowledge of the

sideband ratio (SBR), which is the gain ratio between the upper and lower sideband

frequencies. For an ideal double-sideband receiver the SBR is equal to one, but in

practice the receiver response in the upper sideband may be different from that in

the lower sideband. Therefore it is very important to know the SBR at different LO

frequencies to be able to recover, from a measured spectrum, the correct relative

12

intensity of the various spectral lines. This is because even if a spectral line is

observed in one sideband, atmospheric noise from the other sideband is

superimposed on the spectrum.

At the moment the spectroscopic techniques developed for sub-millimeter analysis

to determine the SBR (such as Martin Pupplet interferometry [8] and Gas cell

technique [9]) rely on a calibrated filter system and are rather complicated.

1.4 Thesis goal and outline

In this thesis sideband ratio measurements of a sub-millimeter receiver in the 600-

720 GHz band (ALMA Band 9) using a Michelson interferometer as input filter are

presented. The main goal is the investigation of the relation between the direct and

heterodyne mode to determine if the simple direct detection method is a reliable

predictor of the SBR of double sideband mixers.

Chapter 2 contains an overview of the theory of superconductivity and the

description of the type of superconducting mixers used in this thesis.

In chapter 3 the Michelson interferometer and the complete instrumental setup are

described in detail.

The standard technique used to calculate the SBR for sideband-separating mixers is

rather complicated [10]. Here we propose a new method to determine the SBR of

sideband-separating mixers using heterodyne spectra measured with a Michelson

interferometer. The experimental setup is similar to the one used during the

measurements with double sideband mixers and it is also described in detail in

chapter 3.

The results obtained with double sideband mixers are reported in chapter 4, and

those with sideband-separating mixers in chapter 5.

13

Chapter 2

SUPERCONDUCTOR-BASED MIXERS

2.1 Phenomenology of superconductivity

Superconductivity was discovered in 1911 by Heike Kamerlingh Onnes and

coworkers as they were conducting experiments on the resistance of metallic

samples at cryogenic temperatures. The superconducting phase of a material is

characterized by the absence of electrical resistivity below a critical temperature Tc

(fig. 2.1) [11]. This phase also behaves as a perfect diamagnet: an external magnetic

field, below a certain strength, is not able to penetrate inside the superconductor.

Moreover, if a magnetic field is present at the moment that T goes below Tc, the

field is expelled from the superconductor. This is known as the Meissner effect and

is represented in figure 2.2. On the contrary, above Tc most superconducting

materials behave as normal conductors: they display a non-zero resistivity, and an

applied magnetic field penetrates uniformly through the material generating a

paramagnetic response.

Figure 2.1: Behaviour of the resistivity ρ as a function of temperature T for a normal metal

(blue line) and a superconductor (red line).

14

Figure 2.2: Magnetic field distribution in the normal state (T>Tc) and in the

superconducting state (T<Tc) of a superconductor in the presence of a

homogeneous external magnetic field.

Bardeen, Cooper and Schrieffer developed in 1957 the theory of conventional

superconductors (known as BCS theory [12]). Electrons in solids usually repel one

another due to the mutual Coulomb interaction. However, in the presence of other

interactions, the net coupling between two electrons in a metal can be attractive. As

an electron moves in a metallic crystal, it attracts the positive lattice ions, generating

a lattice distortion (in quantum mechanical terms, it emits a phonon). This lattice

distortion creates an excess local positive charge which persists also after the passing

of the electron (as the speed of atomic ions is much lower than that of electrons),

and is thus able to attract a second electron (which, in quantum terms, absorbs the

emitted phonon).

This results in a “retarded” attractive potential mediated by the electron-phonon

interaction. Although this effective attraction is weak, it is nonetheless effective as

the electrons avoid meeting directly and thus do not feel their full mutual Coulomb

repulsion. The attractive potential leads to the formation of electron pairs with

opposite momenta and opposite spin (called “Cooper pairs”), which behave as

bosons and are thus able to collapse in a Bose-Einstein-like condensate at low

temperature. The BCS theory describes superconductivity as the formation of a

macroscopic, coherent condensate of Cooper pairs. Since the electrons are bonded

in Cooper pairs, a finite amount of energy (∆) is needed to break them in two

15

independent electrons. Thus an energy gap 2∆ in the density of states is present (fig.

2.3) which is usually of the order of a few meV and which decreases with increasing

temperature and vanishes at the critical temperature, when superconductivity ceases

and the normal metallic state sets in. The presence of this gap also accounts for the

lack of resistivity. The resistivity is zero because Cooper pairs have no available

states close in energy to scatter into, contrary to the case of a normal metal where

electron states arbitrarily close to the Fermi level are available.

Figure 2.3: Electron densities of states for a metal at T>0 (left), and a BCS superconductor

at T=0 (right).

2.2 Superconductor-Insulator-Superconductor Junction

A sequence of superconductor-insulator-superconductor layers forms a so-called

SIS junction. For a thick insulating layer no current can flow between the

superconductor electrodes, even if the applied potential across the junction is not

zero. But if the insulator is very thin (few nanometers), a current can start to flow

because of the quantum mechanical tunnelling effect. At high temperatures where

the superconducting material is in the normal state, single electrons tunnel through

the barrier. In the superconducting state, in the absence of pair-breaking excitations,

only Cooper pairs can tunnel through the junction. The flow of Cooper pairs

through a barrier or inside the superconductor is called “supercurrent”.

Figure 2.4 shows the band diagram of a SIS device when it is cooled below the

critical temperature of the two superconductors.

16

Figure 2.4: Geometry and energy band diagram of a SIS junction. The energy gap 2∆ forms

when the junction is cooled below the critical temperature.

If there is no voltage difference across the barrier, the occupied states of both

superconductors are aligned and only Cooper pairs from either side of the barrier

can tunnel through it. This effect, known as DC Josephson effect, produces a

current without generating a potential across the barrier; if the two superconductors

are connected electrically at each end of the junction, such supercurrent persists

forever. The maximum possible supercurrent (called the critical current of the

junction IC) is limited by the tunnelling probability and by the density of the Cooper

pairs present in the junction. With a fixed voltage across the junction, the phase will

vary linearly with time and the supercurrent will be an alternate (AC) current. This is

known as AC Josephson effect.

Both types of Josephson effects are related to the existence of a difference in the

phase of the wavefunction of the Cooper-pair condensate across the junction. The

tunnelling supercurrent can be suppressed by the application of a magnetic field

across the barrier, which alters the phase difference between the condensate

wavefunction at the different ends of the junction. In this way, a carefully chosen

magnetic field quenches supercurrent and the only current that is flowing through

the junction is due to “quasiparticles”1 (“leakage current”), which are unpaired

1 The ground state of the system is the condensate of Cooper pairs. In general, quasiparticles are defined as elementary excitations of a system which have particle-like properties (i.e., a defined momentum, energy, mass, charge, etc.).

17

electrons, or to tunnelling of Cooper pairs into separate quasiparticle states. The

latter process can only happen if there is enough potential energy available to break

the Cooper pair (i.e. 2∆).

A SIS junction in which the supercurrent is suppressed exhibits a non-linear

current-voltage (I-V) behaviour as shown in figure 2.5. The main features of the I-V

curve of a SIS junction are:

• At voltages │V│< VG (where VG is the gap voltage, given by 2∆/e) the

occupied electronic states in the two superconductors overlap in energy and

there is no significant quasiparticle tunnelling through the barrier, if no other

sources of excitation are available (see section 2.2.1).

• At a bias voltage │V│= 2∆/e the occupied states of one superconductor are

aligned with the empty states in the other superconductor and Cooper pairs

of the former can tunnel into quasiparticle states in the latter. The density of

states diverges at the edges of the superconducting energy gaps, which causes

a step-function increase in the intensity of the bias current.

• Increasing the bias voltage above the gap voltage, the SIS junction exhibits a

metallic-like behaviour with a linear increase of the current with the voltage.

Figure 2.5: Current-voltage curve of SIS junction. The dashed line is the current-voltage curve

when the SIS junction is pumped with a radiation in the sub-mm range.

18

2.2.1 Photon-assisted tunnelling in a SIS junction

If a radiation of frequency ν is applied across the SIS junction (that is, the SIS

junction is pumped), photons can be absorbed by quasi-particles or by Cooper pairs.

If one photon at a time is absorbed, the energy of the quasi-particles raises by the

energy of a quantum hν (or nhν if n photons are absorbed at a time). If the bias

voltage is in the range (2∆-nhν)/e <│V│< 2∆/e, then the quasiparticles or Cooper

pairs can tunnel, upon light absorption, to the allowed empty states across the

junction (figure 2.4). This process is named photon-assisted tunnelling and it gives

rise to sharp steps in the current at voltage intervals of hν/e. The dashed line in

figure 2.5 shows the current-voltage characteristics of a SIS junction pumped with a

sub-millimeter wavelength radiation. The number of electrons tunnelling through

the barrier is related to the number of photons absorbed and thus to the power of

the radiation source.

2.2.2 SIS junction as mixing element

A common feature of mixer components is their non-linear I-V curve in a certain

range of voltages. As shown in the introduction, this non-linearity is responsible for

the frequency mixing (creation of frequency combinations as the sum or difference

of incoming frequencies) when two signals with different frequencies are applied.

The frequency component with the difference of these two frequencies (IF) is the

crucial one for heterodyne mixing. As visible in figure 2.5, SIS junctions show a

non-linear I-V behaviour for bias voltage below the gap voltage. The possibility to

use a SIS junction as a mixer is a consequence of this non-linearity and of the

quantum-mechanical nature of the photon-assisted tunnelling [13]. When a time-

varying voltage V(t) is applied to a SIS junction of which one side is connected to

ground, the quasiparticle energy eigenstates on the ungrounded side of the barrier

are modulated by a phase factor f(t). From this factor, the current that tunnels

through the device can be written as:

19

∑∞

−∞=

=n

tjnenIti

ω)(

2

1)(

(2.1) where I(n) are complex currents (current phasors).

Similarly, when the applied voltage comprises two periodic wave forms that have

different frequency ω1 and ω2, the tunnelling current is given by:

∑ ∑∞

−∞=

∞

−∞=

+=

m n

nmjenmIti

)( 21),(2

1)(

ωω (2.2)

From this equation it can be seen that the tunnelling current contains several terms

at frequencies which are linear combinations with integer coefficients of ω1 and ω2.

One of this combination will be ω1-ω2, which corresponds to the intermediate

frequency.

2.2.3 Double sideband (DSB) mixers

In a DSB mixer both the upper and lower sidebands are detected. The unwanted

sideband can be suppressed using a filter in front of the receiver resulting in a so-

called single sideband (SSB) mixer.

Figure 2.6 shows a block diagram of a so-called single-ended or double-sideband

sub-millimeter SIS mixer. The input to the mixer is in this case an (quasi-)optical

beam formed by the coupling between the LO and antenna signals, which can be

performed through beamsplitter, Martin-Pupplett or pure waveguide techniques.

This is collected by a focussing optical element and coupled into the SIS junction,

producing an intermediate frequency output which is then amplified and detected.

Two common ways to collect the signal and focus it onto the detector are a horn-

waveguide probe or a lens-antenna. In the method using the lens-antenna system

quasi-optical coupling is employed. A dielectric lens is used to focus the radiation on

a planar antenna mounted on the back of the lens (fig. 2.7). An advantage of this

technique is the relative ease of fabrication.

20

In the waveguide method, the input optical beam is focussed into a horn antenna

and passed through a waveguide. The SIS chip is mounted in a substrate channel

that crosses this waveguide. A transmission line and impedance transformer then

couples the radiation from the waveguide probe to the SIS junction, in which the

RF and LO signals are mixed (fig. 2.8). Despite the more laborious fabrication (it

becomes more difficult to fabricate as the wavelength decreases), this coupling

system is largely used for sub-mm band detection because of the good beam quality.

This method of coupling is used in the ALMA Band 9 receivers studied in this

thesis. The DSB mixers designed for ALMA Band 9 work in the frequency range

from 600 to 720 GHz, with an IF range of 4 to 12 GHz.

Figure 2.6: Block diagram of a typical sub-millimeter SIS mixer.

Figure 2.7: Lens-antenna system.

21

Figure 2.8: DSB mixer used in ALMA Band 9. Upper panel: expanded sketch of the

assembly of horn and mixer. The setup comprises the horn (A), the mixer

backpiece (B) and the IF output channel (C).

Lower panel: structure of the mixer and detail of the SIS junction inside it.

2.2.4 Sideband-separating (2SB) mixers

In 2SB mixers, the unwanted image channel is not filtered out but is separated from

the signal channel. The configuration of the 2SB mixer designed as a possible

upgrade for ALMA Band 9 is shown in figure 2.9. With the use of a device called a

quadrature hybrid, the input RF signal is separated in two parts with a phase

difference of 90°. These two signals are then independently combined with an in-

phase LO signal and each one is fed into a different SIS junction (DSB). The down-

converted signals are sent into another quadrature hybrid. In one output of this

hybrid the LSB signal cancels out, while in the other the USB signal cancels out.

Thus the USB and LSB signals are obtained separately.

To better explain the working principle of the 2SB mixer, let us consider in detail

the scheme of figure 2.9a (see Ref. 14 for similar calculations). As the incoming RF

signal passes through the first hybrid, it is split in two parts which have a relative

22

phase of 90°. We are interested in the components of the RF signal corresponding

to the USB and LSB at frequencies ωU and ωL, respectively. These two components

arrive with different phases at points A and B in figure 2.9a. At point A, the

components have the following time dependence:

)cos(

)cos(

tLSB

tUSB

L

u

ω

ω

∝

∝ (2.1)

while at point B one has:

)90cos(

)90cos(

°+∝

°+∝

tLSB

tUSB

L

U

ω

ω (2.2)

The USB and LSB signals are then down-converted by the DSB mixer with the

insertion of the LO signal at frequency ωLO. At point C the signals are then:

[ ] [ ]

[ ] [ ]ttttttLSB

ttttttUSB

IFLOIFIFLOLOLLO

IFLOIFIFLOLOULO

)2(cos2

1)cos(

2

1)(cos)cos()cos()cos(

)2(cos2

1)cos(

2

1)(cos)cos()cos()cos(

ωωωωωωωω

ωωωωωωωω

−+=−=∝

++=+=∝ (2.3)

Only the first term after the last equal sign is considered. At point D, in a similar

manner, the signals are:

[ ] [ ]

[ ] [ ]°−+°−=°+−∝

°+++°+=°++∝

+ 90)2(cos2

1)90cos(

2

190)(cos)cos(

90)2(cos2

1)90cos(

2

190)(cos)cos(

ttttLSB

ttttUSB

IFLOIFIFLOLO

IFLOIFIFLOLO

ωωωωωω

ωωωωωω (2.4)

At point E after the second hybrid, the sidebands from point C are combined with

the sidebands at point D delayed by 90°. Thus, the two USB signals are combined

with a 180° phase difference (i.e., they are totally out of phase) and therefore cancel

out, whereas the LSB signals arrive in phase and are added up:

[ ]LSBIF

LSB

IF

USB

IF

LSB

IF

USB

IF ttttt )cos()cos(2

1)180cos(

2

1)cos(

2

1)cos(

2

1ωωωωω ⇒

+

°++

+

(2.5)

23

Figure 2.9: a) scheme of the 2SB configuration for ALMA Band 9, b) three-dimensional

scheme of the 2SB mixer core (after Ref. 14), c) picture of the 2SB mixer for

ALMA Band 9.

In the same way, at point F the LSB signals arrive with a 180° phase difference and

cancel each other, while the USB components are combined in-phase:

LSB

IF

USB

IF

LSB

IF

USB

IF tttt

°−+

°++

°++

°+ )90cos(

2

1)90cos(

2

1)90cos(

2

1)90cos(

2

1ωωωω

[ ]

USBIFt )90cos( °+⇒ ω (2.6) A difficulty with the 2SB mixers are the very small dimensions of some components

required at high frequencies. Nevertheless the performance obtained with this

design of the mixer is very good and satisfies the ALMA specifications [15].

A reason to prefer 2SB mixers instead of DSB mixers for high resolution spectral

line observations in radio astronomy is their enhanced sensitivity. As the sensitivity

24

of SIS mixers approaches its fundamental limit, the output noise of a DSB receiver

is degraded by the atmospheric noise coming into the system through the image

band. Even if there is no RF frequency component in the image sideband, the

atmospheric noise in this sideband cannot be removed and contributes to the total

IF output, thereby affecting the overall sensitivity.

25

Chapter 3

INSTRUMENTAL SETUP

3.1. The Fourier Transform Spectrometer: general description

The Fourier transform spectrometer (FTS) is a Michelson interferometer with a

movable mirror. A block diagram of the FTS is shown in figure 3.1. The light from

the source is split in two beams by a beamsplitter. One beam is reflected off a fixed

mirror and one off a moving mirror which introduces a variable time delay. After

the reflection from the mirrors, the beams are recombined coherently at the

beamsplitter and detected. The procedure is repeated for different time delays,

corresponding to different positions of the moving mirror.

The intensity I(∆) of the recombined beam, as a function of the path difference ∆, is

the Fourier transform of the product of the spectral distribution of the source, the

transmission of the optical medium (air, vacuum) and the spectral response of the

detector, as shown below. This means that an FTS can be used either to study the

emission from an unknown source (if the detector is broad-band), or the response

of an unknown detector (if the source is broad-band), or the transmission of the

medium (e.g. filters or window materials) if both are broad-band.

In general, if the light source is monochromatic and is split in two beams of equal

intensity, the electric field component of the recombined beam in the interferometer

can be written as:

)1(2 0

∆+= ikti

T eeEEω (3.1)

where ∆=2x is twice the distance x (see fig. 3.1) of the movable mirror from the

zero path difference position, and k=2π/λ [16].

The intensity of the recombined beam can be written as:

[ ])cos(12

0

2∆+== kEEI T . (3.2)

26

Figure 3.1: Block diagram of the FTS, x is the distance of the movable mirror from the zero

path difference position.

If the light source is not monochromatic, but has a spectral distribution S(k), then

the intensity of the light from the interferometer can be found by adding the

intensities at different wavelengths:

[ ]∫ ∫∫∞ ∞∞

∆+=∆+==∆

0 0

2

0

2

0

2

0)cos()()cos(1)()()( dkkkSIEdkkkSEdkkSEI rT (3.3)

where ∫∞

=0

)( dkkSI r is the reference beam.

The integral in the right-hand equation 3.3 is the Fourier cosine transform of S(k).

Hence, if we have an interferogram ∫∞

∆=∆Φ0

)cos()()( dkkkS , the spectral

distribution may be calculated from the Fourier formula considering that Φ(-∆)=

Φ(∆):

∫∞

∆∆∆Φ=0

)cos()(2

)( dkkSπ

(3.4)

Since cos(k∆) is an even function, the interferogram should be symmetrical about

the white light fringe (∆=0) for a perfectly aligned instrument.

27

Figure 3.2: Example of interferogram (above) with the correspondent frequency spectra

(below) measured by the Michelson interferometer described in this thesis.

An example of interferogram, with the correspondent frequency spectrum,

measured by the Michelson interferometer is shown in figure 3.2.

Because the resolution of an FTS increases with increasing optical path difference,

the maximum spectral resolution, max2x

c=∆ν , is achieved by using the entire

distance to measure one side of the interferogram. However, in order to maximize

the signal-to-noise ratio and to avoid problems with cutting of the interferogram,

both sides of the interferogram are usually measured. The maximum frequency of

the spectral distribution is given by s

c

4max =ν , where s is the step size of the movable

mirror.

The orientation of the mirrors is a critical point: the movable and the fixed mirrors

must be exactly perpendicular with respect to the incident beam. A misalignment

can produce a reduction in the spectral amplitude and cause asymmetries in the

interferogram [17]. For alignment purposes, the source can be replaced by a laser

and the detector by a screen.

28

3.2. Instrumental setup for double sideband mixers

A block diagram and a picture of the instrumental setup used during the experiment

with the double sideband mixers is shown in figure 3.3. The source consists of a

glowbar lamp (black body source) with a chopper in front of it for lock-in

measurements (see section 3.2.2). The first part of the setup consists of a Michelson

interferometer described in section 3.1.

Figure 3.3: Upper panel: scheme of the heterodyne detection setup used with the double

sideband mixers. BS1 is the beamsplitter of the Michelson interferometer. The

beam from BS1 is coupled with the signal from the LO through beamsplitter

BS2. L and G1 are respectively a lens and a grid used to focus and adjust the

amplitude of the LO signal. The grid G2 is used to reduce the standing waves

from the LO as discussed in the text. The rejected beams are sent to two

absorber plates (A). Lower panel: picture of the instrumental setup.

29

In the Michelson interferometer used during this thesis, the maximum spectral

resolution achievable is 750 MHz, corresponding to a difference in path length of

400 mm (or a maximum mirror travel of 200 mm).

When the receiver is used in direct detection mode, the recombined beam is sent to

a parabolic mirror (a description of the mirror design is presented in section 3.2.3)

which focuses it into the cryostat where the receiver is located. In this way it is

possible to obtain the frequency response of the receiver. In heterodyne detection

mode, the beam from the parabolic mirror is coupled with the signal from the local

oscillator using the beamsplitter BS2. The working principle of the LO is presented

in section 3.2.1.

Due to imperfections in the coupling between the incoming signal and the horn

receiver, some LO signal can be reflected back into the Michelson interferometer

forming standing waves. Since the LO signal is polarized, introducing a new grid

(G2) suitably oriented can reduce these standing waves by acting as an adjustable

attenuator.

3.2.1 Local oscillator

The local oscillator is an important part of the heterodyne receiver. It is a device

used to generate a signal which is mixed against the signal of interest to convert it

down to a lower frequency. The LO signal must be coherent and with a high

enough power level (several µW) in order to adequately pump the mixer. The band

coverage, the tuneability and the spectral purity of the LO are also important

parameters of a heterodyne system. For our experiments we used a multiplier chain

local oscillator. In this device, harmonics generators are used to convert a low

frequency signal (generated by a YIG oscillator or a signal generator) to the desired

frequency range. In our case the low frequency range was between 16 and 20 GHz,

and the total multiplication factor was 36 (two doublers and two triplers in series),

producing an LO with a range of about 600 to 720 GHz. This corresponds to

ALMA Band 9. Figure 3.4 shows a picture of the LO used for the experiments.

30

Figure 3.4: Picture of the local oscillator used during the experiments produced by NRAO.

3.2.2 Lock-in amplifier

The lock-in amplifier is used to measure the amplitude and phase of signals buried

in noise. The principle behind a lock-in amplifier is that the quantity to be measured

is modulated and the response is synchronously detected with a very narrow

bandwidth around the carrier. In this way both white noise (because of the narrow

bandwidth) and 1/f noise (because we stay far away from DC) are greatly reduced.

For measurements in which light is detected, the device used to modulate the signal

is usually a chopper. In essence a lock-in amplifier takes the input signal (that

corresponds to the detector output signal), multiplies it by the reference signal

(provided by the chopper) and integrates it over a specified time. The resulting

signal is a DC signal in which the contribution from any signal that is not at the

same frequency of the chopper is reduced practically at zero. The integration time of

the lock-in is determined in order to obtain the best signal to noise ratio, while

keeping the measurement time within reasonable limits. To get sufficient accuracy,

the time during which the mirror is kept at a fixed position between two consecutive

steps should be at least three times the integration time constant.

3.2.3 Design of the parabolic mirrors

During our experiments we made use of a glowbar lamp as source. Because the

beam from the lamp is convergent, a mirror is needed to convert it into

31

approximately a parallel beam in order to feed it into the Michelson interferometer.

To this purpose, we have designed a convex parabolic mirror. The beam aperture of

the lamp, which already includes an elliptical mirror, is 80 mm. The designed mirror

has a focal length of 190 mm and dimensions of 70x70x70 mm. A sketch of the

design is shown in figure 3.5.

We have also designed a concave parabolic mirror which couples the beam from the

Michelson interferometer into the horn and then into the SIS junction. The horn

has a f-number which is related to the diverging angle of its beam. The aperture (a)

of the ALMA horn is about f/3, which means that a/f = 1/3 (see fig. 3.5). As a

consequence, because the width of the beam coming from the convex parabolic

mirror is approximately 50 mm, the focal length of the mirror should be 150 mm.

Due to diffraction effects because of the finite beam size, as well as imperfections of

the convex mirror, the beam can be not perfectly parallel. To ensure that the whole

beam coming from the convex parabolic mirror is reflected to the horn, the

dimensions of the mirror (100x100x100 mm) are oversized (see fig. 3.5).

Figure 3.5: Left: sketch of the design of the convex parabolic mirror and the concave

parabolic mirror. The converging beam from the lamp is converted into a parallel

beam using a convex mirror and then it is focused into the detector using a

concave mirror (the dimensions are in mm). Right: picture of the mirrors. The

convex parabolic mirror is mounted on the support of the glowbar lamp.

32

3.3. Instrumental setup for sideband-separating mixers

A block diagram and a picture of the instrumental setup used for the experiments

with the sideband separating mixers are shown in figure 3.6.

Figure 3.6: Upper panel: scheme of the heterodyne detection setup used with the sideband

separating mixers. BS1 is the beamsplitter of the Michelson interferometer. The

beam from BS1 is inserted into the cryostat through the window W1. The beam

from the LO source is inserted into the cryostat through the window W2. The

use of the beamsplitter BS2 allows the alignment of the LO beam without

moving the cryostat. L and G1 are respectively a lens and a grid used to focalize

and change the amplitude of the LO signal. Lower panel: picture of the

instrumental setup.

33

The difference between this setup and the setup used for the double sideband

mixers reflects the different way of operation of the two types of mixers. The

cryostat used with the 2SB mixer is equipped with two windows (W1 and W2) for

the separate insertion of the input signal and the LO signal, because the mixing

between the two signals is done inside the mixer block in the cryostat. Hence the

signal from the LO is sent through the window W2 using the beamsplitter BS2 as a

simple mirror. The LO signal is not sent directly to the cryostat because in this way

it is possible to align the LO beam without moving the cryostat (which is previously

aligned with the reference beam inserted through the window W1). Since in this case

we use a sideband separating mixer, we have two distinct output signals, one

corresponding to the upper side band and the other to the lower sideband. Both

signals are detected independently (by using two separate lock-in amplifiers locked

to the same reference) in the same way described for the double sideband mixers

(see section 3.2).

34

35

Chapter 4

SIDEBAND RATIO OF DOUBLE SIDEBAND

MIXERS

The purpose of the experiments presented in this chapter is the study of the relation

between the direct and heterodyne detection modes to determine if the simple direct

detection method is a reliable predictor of the sideband ratio of double sideband

mixers.

4.1 Detection with double sideband mixers

The measurements described in this chapter were performed in both direct and

heterodyne detection modes using two different superconductor-insulator-

superconductor (SIS) junctions, hereafter called mixer 1 and mixer 2. Both junctions

are designed to operate in the 600−720 GHz band.

4.1.1 Direct detection measurements

For both direct and heterodyne detection mode (see section 3.2 for the description

of the instrumental setup), the same cryostat filled with liquid helium (4K) can be

used. After the filling of the cryostat the junction is checked by measuring its I-V

curve. The measured curves for mixer 1 and 2 are reported in figure 4.1. Josephson

currents are suppressed with the application of a magnetic field. For both detection

modes, the bias voltage is set to the voltage just before the onset of the jump to

normal resistance, namely around 2.5 V (blue points in fig. 4.1).

36

Figure 4.1: I-V curve of mixer 1(right) and mixer 2(left). The red line corresponds to the I-

V curve with the application of a magnetic field to suppress the Josephson effects

visible in the black line. The blue point corresponds to the set of the bias voltage

(2.5 mV).

For a good experiment the mixer beam should be carefully aligned with the beam

from the Michelson interferometer. This is done by manually moving the cryostat in

order to find the point at which the intensity of the detected signal is maximum.

When the right alignment is found, the response of the mixer can be measured over

the entire frequency range using the direct detection mode.

This procedure is used to acquire the direct response of mixers 1 and 2. For the

detection of the direct spectrum in mixer 1 a maximum length path difference of the

FTS of 16 mm (equal to double of the maximum mirror travel) is used, which gives

a resolution of 18.75 GHz; for mixer 2 the resolution is improved to about 9 GHz

by increasing the length path difference to 32 mm. This also results in an increase of

number of points of the direct spectrum, which allows a better estimate of the area

under the curve for the evaluation of the SBR in direct mode and a better signal to

noise ratio. The results are shown in figure 4.2. It can be seen from this figure that

the detectors are lightly high-tuned with respect to the frequency band of interest

(between 600 and 720 GHz). This does not affect our experiments. The fact that the

maximum output in the response of mixer 1 differs from that of mixer 2 is not

necessarily a feature of the mixers, but may also depends on how precisely the

alignment of the system is done and the precise temperature of the glowbar.

37

Figure 4.2: Direct response spectrum of mixer 1 and mixer 2.

4.1.2 Heterodyne detection measurements: mixer 1

The configuration for the heterodyne detection is the same as for direct detection

but with the insertion of the LO as shown in figure 3.3. No further alignment of the

cryostat was performed after the acquisition of the direct spectrum. Before

measuring, the LO must be aligned in order to optimize the coupling of the LO

signal with the reference signal. First the LO is aligned using a mirror in place of the

beamsplitter BS2. Moving the LO source, the mirror, the lens and the grid in front

of the LO source, it is possible to find a position in which the pumping level of the

junction is maximum. This position can be further adjusted using the beamsplitter

BS2. During heterodyne detection, the LO frequency is varied from 597.6 GHz to

720 GHz in steps of 3.6 GHz. A spectrum is taken for every LO frequency. For

these measurements we have used a path length difference of 300 mm in order to

have a spectral resolution of 1 GHz. Examples of heterodyne spectra acquired with

different LO frequencies are shown in figure 4.3. In each spectrum two peaks can

be clearly distinguished around the LO frequency. These correspond to the LSB and

USB bands.

38

Figure 4.3: Some heterodyne detection spectra at different LO frequencies for mixer 1 as

measured.

Figure 4.4: Comparison between the direct response spectrum (thick grey line) and some

heterodyne detection spectra at different LO frequencies (thin lines) with mixer

1. The spectra are normalized so that the intensity of one of the two sidebands

peaks coincides with the intensity of the direct response at the same frequencies.

A third peak is visible between the two main features at exactly the LO frequency.

This peak is originated by the standing waves from the LO that are reflected back in

the interferometer, causing periodic variations in the pumping level. This means that

this configuration of the setup was not optimal.

39

Since the gain of the mixer and the LO power, and possibly the coupling, is not the

same for measurements at different LO frequencies, each of the heterodyne spectra

can be normalized so that the intensity of one of the two sideband peaks coincides

with the intensity of the direct response at the same frequency. This normalization

does not compromise the SBR results, because we are interested in the peak ratio.

With this procedure we can directly compare the full spectrum obtained in the direct

detection mode with the heterodyne spectra at different LO frequencies. As it can

be seen in figure 4.4, the intensity ratio between the upper and lower sideband peaks

follows approximately the intensity profile of the direct spectrum.

4.1.3 Heterodyne detection measurements: mixer 2

From the heterodyne spectra detected with the mixer 1 some problems in the

instrumental setup are present. Due to imperfections in the coupling between the

incoming signal and the horn receiver, part of the LO signal can be reflected back

into the Michelson interferometer forming standing waves. The insertion of grid G2

(see figure 3.3) almost completely solves this problem. Another problem is the low

pumping level of the LO at certain frequencies. An example of this is the lack of

results between 620 and 640 GHz: the pumping level was not enough to generate a

signal response at these frequencies. To overcome this difficulty, we have split the

frequency range in two subranges, a low-frequency range from 590.4 GHz to 662.4

GHz and a high-frequency range from 666.0 GHz to 720 GHz. For each range we

changed the intensity of the LO beam by rotating the grid G1, in order to optimize

the pumping level. We performed a second set of measurements, with mixer 2,

where these two changes were implemented. The results are shown in Fig. 4.5. It is

evident that the intensity of the central peak in each heterodyne spectrum is lower

and we covered all the frequency range with the heterodyne spectra.

Also with this new configuration of the instrumental setup, the spectra taken in the

heterodyne mode follow quite well the direct response of the mixer 2 (fig. 4.6). The

40

same normalization procedure described in the previous subsection was employed

here.

Figure 4.5: Some heterodyne detection spectra, as measured, at different LO frequencies for

mixer 2 and with the insertion of the grid G2 in the instrumental setup.

Figure 4.6: Comparison between the direct response spectrum (thick grey line) and some

heterodyne detection spectra at different LO frequencies (thin lines) with mixer 2

and with the insertion of the grid G2in the instrumental setup.

41

4.2 Sideband ratio calculation

4.2.1 SBR results

The sideband ratio (SBR) is calculated for each LO frequency as the ratio between

the integrals of the USB and LSB peaks (I1 and I2 respectively in figure 4.7a). This is

then compared with the SBR estimated from the direct detection response, defined

as the ratio between the integrals of the direct-detection curve in the same frequency

ranges of the USB and LSB peaks, respectively (I′1 and I′2 in figure 4.7b).

Figure 4.7: a) Area below the USB (I1) and the LSB (I2) peaks in a heterodyne spectra.

The SBR is the ratio between these two areas, calculated as numerical integrals,

(I1/I2). b) Areas below the direct detection curve in the same frequency range of

I1 and I2. The SBR is defined as the area (integral) ratio I′1/I′2.

42

Figure 4.8: Comparison between the measured SBR in heterodyne mode (ratio between the

integrals of the USB and LSB peaks) and the estimated SBR from the direct

detection response (ratio between the integrals of the direct detection curve in the

same frequencies range of the USB and LSB peaks): a) with mixer 1, b) with

mixer 2.

The result of this comparison for mixer 1 and 2 is reported in figure 4.8. Each

estimate is accompanied by the corresponding error bar (see Appendix A for the

error calculation).

43

4.2.2 Discussion of the results

It can be noted that for both detection modes and for both mixers, the SBR is in the

expected range, i.e., between 0.5 and 2.0 [18] (corresponding to ± 3dB) over the

whole frequency range.

From the results obtained with mixer 1 (see fig. 4.8a) it is seen that there are some

differences between the SBR in the two different detections modes at frequencies

below 640 GHz. The main reason for this is the presence of standing waves coming

from the LO source causing the pumping level to change with the position of the

moving mirror in the Michelson interferometer. This effect is more pronounced at

low frequencies since the coupling between the horn and the incoming signal is also

lower at these frequencies, so that the mixer is under-pumped and it is more

susceptible to LO power variations. With the changes in the instrumental setup, as

discussed in the section 4.1.3, and using mixer 2, a better agreement is found

between the two SBR estimates. The two curves in figure 4.8b have qualitatively the

same shape. However, also when the noise error bars are considered, there is a

quantitative discrepancy in the frequency range between 615 and 660 GHz. This

discrepancy is always less than 16% as shown in figure 4.9. Hence the direct

detection can be used, at least in the case of the mixer 2, to predict the SBR if an

accuracy of 16% is sufficient.

Figure 4.9: Discrepancy (in %) between the SBR calculated from the heterodyne and direct

detection for mixer 2.

44

Figure 4.10: Effective splitting of the i-th quasiparticle level (of unperturbed energy Ei) of an

ungrounded superconductor in the presence of a AC applied voltage. A

microwave filed impinging on the superconductor acts as an AC voltage.

More statistics is needed, using different mixers, to give a more reliable estimate for

the discrepancy. We estimate however that an accuracy of the order of 20% is a

reasonable assumption.

The main reason for the discrepancy is presumably that the heterodyne and direct

detection modes are physically different. In particular, the heterodyne detection is

based on the non-linear mixing of two signals, while in the direct mode only the

source signal is present. The discrepancy may also be caused by other factors, such

as the embedding impedance and the mixer impedance, both of which depend on

the frequency and pumping level (close to zero in the case of the direct detection).

In the following, we analyze in particular the difference between the output DC

current in the direct and heterodyne modes [19].

A (weak) microwave field impinging on a junction in which one of the two

superconductors is grounded, acts as an AC voltage applied to the ungrounded side

of the junction. This sinusoidal perturbation effectively splits each quasiparticle level

in the superconductor into a superposition of (virtual) levels separated by multiples

of the photon energy ħω (see fig. 4.10), occurring with a probability amplitude

Jn(eVω/ħω), where Jn is the Bessel function of order n. Hence the DC tunnelling

current I0 in the direct detection mode is the sum over the possible tunnelling

processes from the grounded superconductor to each of the resulting levels. This is

given by the expression:

45

∑∞

−∞=

+

=

n

DCne

nVI

eVJVVI

ω

ωω

ω

h

h0

2

00 ),( (4.1)

where V0 and Vω are respectively the DC bias voltage and the AC voltage due to

the microwave source. IDC(V) represents the unpumped DC I-V characteristic of

the junction, so that IDC(V0+nħω/e) is the tunnelling current into the n-th level. In

an analogous way the AC part Iω of the mixer output can be calculated. We are only

interested in the DC part.

In the heterodyne mode we have to consider also the strong signal from the LO.

The IF is so low that the mixing element sees the same high-frequency RF circuit at

the signal and LO frequencies. Therefore the source signal can be considered as a

small differential change in the applied LO waveform, and the generated IF

amplitudes can be considered as small differential changes in the DC current and

voltage. The effect of this signal along with the LO is to produce a small modulation

dI0 of the DC output current I0 (and similarly for the AC component of the output):

ωωω

ω

dVGdVGdVV

IdV

V

IdI 0000

00

0

00 +=

∂

∂+

∂

∂= . (4.2)

The small IF current amplitude is equal to dI0 in the presence of a fixed bias voltage.

The output current in the heterodyne mode is thus:

ωω dV

GG

GGdII

L

LVIF DC

00

00

+== (4.3)

where GL is the output load conductance and G00 and G0ω are the differential

conductances given by:

[ ]{ }∑

∑∞

−∞=

+

∞

−∞=

+−++

=

∂

∂=

+

=

∂

∂=

n

DCDCnn

n

DCn

enVIenVIeV

JeV

Je

V

IG

dV

enVdIeVJ

V

IG

)/(/)1(

)/(

0010

0

0

02

0

000

ωωωωω

ω

ω

ωω

ω

ω

ω

hhhhh

h

h (4.4)

46

In the last expression we have used the definition of G00 and G0ω and the equation

4.1 for I0. It is seen that the output current has a different expression in the two

detection modes, as in the heterodyne mode the IF current is the differential

increment of the direct-mode output current I0 with respect to the modulation of

the LO signal due to the RF signal.

To quantitatively understand the origin of the measured discrepancy between the

direct and heterodyne detection mode, it would be necessary to use a model which

takes into account all the above-mentioned factors, including a numerical simulation

of the expected output currents with equations (4.1) and (4.3).

4.3 Conclusions

Despite a qualitative agreement, a quantitative discrepancy is found between the

SBR calculated from the direct detection and heterodyne detection modes in the

frequency range between 615 and 660 GHz, which can be only partially reduced by

suppressing the LO standing waves in the heterodyne mode. With the improved

setup, the discrepancy between the SBR calculated from the direct and heterodyne

spectra of mixer 2 is always less than 16%. We can reasonably assume that, in

general, the direct detection can be used to predict the SBR if an accuracy of 20% is

sufficient. More experiments are needed to obtain a more reliable estimate for the

SBR discrepancy, accompanied by simulations to understand in detail its causes,

which might be mainly related to the physical non-equivalence between the two

detection techniques and to the different detector response in the two cases.

47

Chapter 5

SIDEBAND RATIO OF SIDEBAND-SEPARATING

MIXERS

In the case of sideband-separating mixers, the sideband ratio is the ratio between the

intensity of the passed band and the rejected band, also called the “image rejection”.

In the ideal case the image rejection is infinite. The standard method to calculate the

image rejection in sideband-separating mixers is rather complicate (see section 5.1),

thus we will try to compare the image rejection of sideband-separating mixers

calculated with such method with the image rejection calculated using heterodyne

spectra measured with a Michelson interferometer.

5.1 Description of the standard method

The standard technique used to calculate the SBR for sideband-separating mixers

involves the injection of a continuous wave (CW) signal of known relative amplitude

into the upper and lower sidebands, and measuring the IF response to each. At

millimeter wavelength, however, it is difficult to determine with sufficient accuracy

the relative amplitude of two low level RF signals. To overcome this problem, it is

possible to measure the SBR of a sideband-separating mixer without knowledge of

the RF signal levels, by measuring the IF response to broadband RF noise sources at

two distinct temperatures [10].

Let us consider a sideband-separating receiver in which only the conversion from

the upper and lower sideband to the IF band is significant (conversion from higher

harmonics is negligible). The scheme of such a receiver is shown in figure 5.1. Gi,j

are the conversion gains from each RF input port to each IF output port.

48

For this receiver the image rejections are:

RUSB=G1U/G1L at the IF port 1, and (5.1)

RLSB=G2L/G2U at the IF port 2. (5.2)

To determine these quantities the following measurements must be performed:

• The IF signals at ports 1 and 2 are measured by applying a CW test signal of

unknown amplitude at the upper sideband port U. The ratio of the powers of

the two outputs is MU=G1U/G2U.

• The IF signals at ports 1 and 2 are measured by applying a CW test signal at

the lower sideband port L. The ratio of the powers of the two outputs is

ML=G2L/G1L.

• By replacing a cold load at the receiver by a hot load, the variation of the

output power at the two IF ports is measured. These changes are defined as:

∆P1=kB∆T(G1U+G1L) and (5.3)

∆P2=kB∆T(G2U+G2L) (5.4)

where ∆T is the noise temperature difference between the two loads. From the

measured quantities MU, ML and MDSB=(∆P1/ ∆P2) it is possible to calculate the

image rejections RUSB and RLSB:

DSBU

DSBLUUSB

MM

MMMR

−

−=

1 and 1−

−=

DSBL

DSBULLSB

MM

MMMR (5.5)

Figure 5.1: Sketch of the sideband-separating receiver. Gi,j are the power gains from each

RF input port (U and L) to each IF port (1 and 2).

49

5.2 Heterodyne detection measurements

The configuration for the heterodyne detection was shown in figure 3.5. The

procedure for the measurements is the same as described for the double sideband

receiver in section 4.1.2. The LO frequency is varied from 612 GHz to 693 GHz in

steps of 9 GHz, and a mirror travel of 150 mm in the Michelson interferometer is

used in order to have a spectral resolution of 1 GHz.

Since the sideband separating mixers have two output signals, two spectra are

measured for every LO frequency: one corresponding to the LSB channel and the

other to the USB channel. These spectra are shown in figure 5.2. In these first set of

measurements some contributions of standing waves at LO frequency are visible.

Figure 5.2: Heterodyne spectra of the LSB channel (above) and of the USB channel (below).

50

These standing waves are presumably an unwanted side effect produced in the

coupling between LO and RF inside the mixer block in the cryostat. Referring to

figure 2.9a, during LO injection some contribution (~1%) of the LO signal is sent

towards the RF port and subsequently in the Michelson path through the window

W1. To check if these standing waves from the LO are the cause of the peak at the

LO frequency in the heterodyne spectra of our 2SB mixer, an approach could be to

put a grid in front of the window W1 and try reducing them by changing the grid

polarization. In general these waves can also be generated from leakage and

reflections of the LO signal in the cryostat or in the mixer block. In both cases an

improvement of the instrumental setup is necessary in order to reduce the central

peak in the heterodyne spectra.

5.3 Image rejection calculations

The calculation of the image rejection (R) is similar to the SBR calculations based on

the heterodyne spectra for the double sideband receivers. If L1 and U1 are

respectively the areas below the LSB and USB peaks of the spectrum measured at

the LSB channel, its image rejection is defined as RLSB=L1/U1 (see fig. 5.3a).

Analogously, the image rejection of the USB channel is defined as RUSB=U2/L2,

where L2 and U2 are respectively the areas below the LSB and USB peaks in the

spectra measured at the USB channel (see fig. 5.3b).

The results of this calculation [15] are shown in figure 5.4 together with the image

rejection calculated with the method described by Kerr in Ref. 7. The error on the

image rejection is obtained for each heterodyne spectrum with the method

described in section 4.3.

51

Figure 5.3: Area below the USB and LSB peaks in a heterodyne spectrum obtained from

the LSB channel (a) of the sideband-separating mixer and from the USB

channel (b). The image rejection for each channel is the ratio between these two

areas (calculated as numerical integrals), namely RLSB=L1/U1 and

RUSB=U2/L2.

52

Figure 5.4: Comparison between the image rejection calculated with the method described by

Kerr and the image rejection calculated from the heterodyne spectra for both the

LSB (above) and the USB (below) channels.

5.4 Conclusions

These first results are quite satisfactory, but there are some differences between the

image rejections calculated with the two methods. This has probably do to the

presence of standing waves at the LO frequency in the signal path. It is therefore

necessary to improve the experimental setup to investigate if the standing waves are

the cause of the discrepancy and if they can be removed.

53

APPENDIX A

SBR ERROR CALCULATION

In the calculation of the error we take into account only the statistical error due to

the finite integration time. To calculate the error associated with the calculated SBR,

we consider an example of detected spectrum in figure A.1. In general, the integral

in the A-B interval can be approximated as:

∫ ∑ ∆≈=B

A i

ii xpdxxfI )(

where pi is the height of the curve at the point xi and f(x) is the curve that

interpolates the points pi. In our case all intervals ∆xi have the same length ∆x. The

absolute error E(pi) associated with each point pi can be assumed to be the same for

every point of the curve. An estimate for this error can be obtained by considering

the signal maximum in the part of the spectrum which is “spectroscopically empty”.

We denote such maximum by Ep in figure A.1.

In general, the combined uncertainty ∆Y of a measurement Y=f(x1,x2,…,xN) is

defined as [20]:

( ) ( )∑ ∑∑= = =

∆∆∂

∂

∂

∂+∆

∂

∂=∆

N

i

N

i

j

N

j

i

ji

i

i

xxx

f

x

fx

x

fY

1 1 1

2

2

22

If the quantities xi are uncorrelated, the term containing the partial derivatives

vanishes. Hence the total (absolute) error EI associated with the value of the integral

can be written as:

( ) nxEpExE pi iiI ∆=∆= ∑ 2)(

The SBR is a ratio between two integrals (I1 and I2), hence the error on the ratio is:

2

2

2

2

1

1

2

2

2

1

21

∆+

∆=

+

=

I

xEn

I

xEnSBR

I

E

I

ESBRE

ppII

SBR

54

Figure A.1: Example of a detected spectrum. pi are the points forming the spectrum, ∆xi the

integration sub-intervals and Ep is the error associated to every point pi.

where n1 and n2 are the number of sub-intervals in which AB is divided. The value

n∆x is the length L of the integration interval AB, which in our case equals to 8

GHz, corresponding to the IF band. Hence the error associated at the calculated

value of the SBR can be written as:

2

22

2

11

⋅

⋅+

⋅

⋅=

nI

EL

nI

ELSBRE

pp

SBR.

55

APPENDIX B

PROGRAMS WITH MATHEMATICA 5.2 1. Calculation of the SBR for double sideband mixers

H∗SBR heterodyne mode∗LSetDirectory@"C:\\DSB\\SBR"D;files= FileNames@"∗.fts"D;result= 8<;intHet1= 8<;intHet2= 8<;ratioDataHet= 8<;c= Length@filesD;ForAi= 0,i≤ Hc−1L, i++;data= Import@files@@iDD,"table"D;n= Dimensions@dataD;s= n@@1DD−8;data= Take@data, −sD;f= Interpolation@dataD;fLO= H16.30+i∗0.10L ∗36∗10^9;

R1= xê.FindRoot@f@xD� 0.003, 8x, fLO−10∗10^9<D;R2= xê.FindRoot@f@xD� 0.003, 8x, fLO−3∗10^9<D;int1= ‡

R1

R2f@xD �x;

intHet1= Append@intHet1,int1D;R3= xê.FindRoot@f@xD� 0.003, 8x, fLO+5∗10^9<D;R4= xê.FindRoot@f@xD� 0.003, 8x, fLO+13∗10^9<D;int2= ‡

R3

R4f@xD �x;

intHet2= Append@intHet2,int2D;ratio= int2êint1;ratioDataHet= Append@ratioDataHet,ratioD;result= Append@result, 8fLO,ratio<D;E;

g1= ListPlot@result, AxesLabel→ 8"Hz",""<,PlotStyle→ PointSize@0.02D, PlotRange→ 80,3<,DisplayFunction→ IdentityD;

Export@"int1HetD33",intHet1,"list"D;Export@"int2HetD33",intHet2,"list"D;Export@"ratioHetD33",ratioDataHet,"list"D;Export@"ratioHetD33.dat",resultD;

56

H∗SBR direct mode∗LSetDirectory@"C\\DSB\\FTS"D;result2= 8<;intDir1= 8<;intDir2= 8<;ratioDataDir= 8<;data2= Import@"ftsD33.fts","table"D;n= Dimensions@data2D;s= n@@1DD−8;data2= Take@data2, −sD;f= Interpolation@data2D;ForAi= 0, i≤ 36,i++;fLO2= H16.30+i∗0.10L∗36∗10^9;

fA = fLO2−12∗109;

fB= fLO2−4∗109;

int1= ‡fA

fBf@xD �x;

intDir1= Append@intDir1, int1D;fC= fLO2+4∗109;

fD= fLO2+12∗109;

int2= ‡fC

fDf@xD �x;

intDir2= Append@intDir2, int2D;ratio2= int2êint1;ratioDataDir= Append@ratioDataDir, ratio2D;result2= Append@result2, 8fLO2,ratio2<D;E;

g2= ListPlot@result2, AxesLabel→ 8"Hz", ""<, PlotStyle→ 8Hue@0.5D,PointSize@0.02D<,PlotRange→ 80, 3<,DisplayFunction→ IdentityD;

g= Show@g1, g2, DisplayFunction→ $DisplayFunctionD;Export@"ratioDirD33.dat", result2D;Export@"int1DirD33",intDir1, "list"D;Export@"int2DirD33",intDir2, "list"D;Export@"ratioDirD33", ratioDataDir, "list"D;

57

2. Calculation of the error associated with the SBR of DSB

H∗Errore direct mode∗LSetDirectory@"C:\\DSB\\FTS"D;data= Import@"ftsDSB.fts", "table"D;n= Dimensions@dataD;s= n@@1DD−9;data= Take@data, −sD;data= Take@data,48D;Err= Max@data@@All,2DDDSetDirectory@"C:\\DSB\\errorData"D;files= FileNames@"∗"D;errorDirect= 8<;int1= Import@files@@1DD, "list"D;int2= Import@files@@3DD, "list"D;ratio= Import@files@@5DD, "list"D;c= Length@int1D;ForAi= 0, i≤ Hc−1L, i++;ErrDir= ratio@@iDD∗$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ikjj H8∗10^9L ∗Err

int1@@iDD y{zz2 +ikjj H8∗10^9L∗Errint2@@iDD y{zz2;

errorDirect= Append@errorDirect, ErrDirD;E;Export@"ErrorDirectD33.dat",errorDirectD;H∗Error Heterodyne mode∗LSetDirectory@"C:\\DSB\\SBR"D;error= 8<;files= FileNames@"∗.fts"D;For@i= 0, i≤ 36,i++;data= Import@files@@iDD, "table"D;

n= Dimensions@dataD;s= n@@1DD−8;data= Take@data, −sD;data2= Take@data, 815, 550<D;Err2= Max@data2@@All,2DDD;error= Append@error, Err2D;D;

int1= Import@files@@2DD, "list"D;int2= Import@files@@4DD, "list"D;ratio= Import@files@@6DD, "list"D;errorHet= 8<;c= Length@int1D;ForAi= 0, i≤ Hc−1L, i++;ErrHet= ratio@@iDD∗&'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''i

kjjjjH8∗10^9L ∗error@@iDD

int1@@iDD∗è!!!!9

y{zzzz2

+ikjjjjH8∗10^9L∗error@@iDD

int2@@iDD∗è!!!!9

y{zzzz2

;

errorHet= Append@errorHet, ErrHetD;E;Export@"ErrorHeterD33.dat",errorHetD;

58

3. Calculation of RUSB and RLSB for sideband-separating mixers

SetDirectory@"C:\\2SB\\data\\heter"D;filesLSB= FileNames@"∗.fts"D;filesUSB= FileNames@"∗.fts2"D;rUSB= 8<;rLSB= 8<;L1data= 8<;U1data= 8<;L2data= 8<;U2data= 8<;rUSBdata= 8<;rLSBdata= 8<;c= Length@filesLSBD;ForAi= 0, i≤ Hc−1L, i++;dataLSB= Import@filesLSB@@iDD, "table"D;dataUSB= Import@filesUSB@@iDD, "table"D;n= Dimensions@dataLSBD;s= n@@1DD−8;dataLSB= Take@dataLSB, −sD;dataUSB= Take@dataUSB, −sD;fLSB= Interpolation@dataLSBD;fUSB= Interpolation@dataUSBD;fLO= H16.75+i∗0.25L ∗36∗10^9;

fA = fLO−8∗10^9;

fB= fLO−4∗10^9;

fC= fLO+4∗10^9;

fD= fLO+8∗10^9;

L1= ‡fA

fBfUSB@xD �x;

U1= ‡fC

fDfUSB@xD �x;

L2= ‡fA

fBfLSB@xD �x;

U2= ‡fC

fDfLSB@xD �x;

R1= U1êL1;R2= L2êU2;L1data= Append@L1data, L1D;U1data= Append@U1data, U1D;L2data= Append@L2data, L2D;U2data= Append@U2data, U2D;rUSB= Append@rUSB, 8fLO, R1<D;rUSBdata= Append@rUSBdata, R1D;rLSB= Append@rLSB, 8fLO, R2<D;rLSBdata= Append@rLSBdata, R2D;E;

Export@"imrejectionUSB.dat", rUSBD;Export@"imrejectionLSB.dat", rLSBD;Export@"L1data", L1data, "list"D;Export@"U1data", U1data, "list"D;Export@"imrejUSBdata", rUSBdata, "list"D;Export@"L2data", L2data, "list"D;Export@"U2data", U2data, "list"D;Export@"imrejLSBdata", rLSBdata, "list"D;

59

4. Calculation of the error associated with RUSB and RLSB H∗error USB∗LSetDirectory@"C:\\2SB\\heterUSB"D;error= 8<;files= FileNames@"∗.fts2"D;For@i= 0, i≤ 9, i++;

data= Import@files@@iDD, "table"D;n= Dimensions@dataD;s= n@@1DD−8;data= Take@data, −sD;data2= Take@data, 815,500<D;Err= Max@data2@@All, 2DDD;error= Append@error, ErrD;D;

SetDirectory@"C:\\2SB\\errorData2SB"D;files= FileNames@"∗"DL1= Import@files@@5DD, "list"D;U1= Import@files@@7DD, "list"D;rUSB= Import@files@@4DD,"list"D;errorUSB= 8<;errorInt= 8<;ForAi= 0, i≤ 9, i++;ErrInt= $%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%H4∗10^9∗error@@iDDL2

4;

errorInt= Append@errorInt,ErrIntD;ErrUSB= rUSB@@iDD∗$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ErrInt2∗

ikjj 1HU1@@iDDL2 +1HL1@@iDDL2 y{zz;

errorUSB= Append@errorUSB,ErrUSBD;E;Export@"ErrorImRejUSB.dat", errorUSBD;

60

H∗error LSB∗LSetDirectory@"C:\\2SB\\heterLSB"D;error= 8<;files2= FileNames@"∗.fts"D;For@i= 0, i≤ 9, i++;

data3= Import@files2@@iDD, "table"D;n= Dimensions@data3D;s= n@@1DD−8;data3= Take@data3, −sD;data4= Take@data3, 815, 500<D;Err= Max@data4@@All, 2DDD;error= Append@error, ErrD;D;

SetDirectory@"C:\\2SB\\errorData2SB"D;files= FileNames@"∗"DL2= Import@files@@6DD, "list"D;U2= Import@files@@8DD, "list"D;rLSB= Import@files@@3DD, "list"D;errorLSB= 8<;errorInt= 8<;ForAi= 0, i≤ 9,i++;ErrInt= $%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%H4∗10^9∗error@@iDDL2

4;

errorInt= Append@errorInt,ErrIntD;ErrLSB= rLSB@@iDD∗$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ErrInt2∗

ikjj 1HU2@@iDDL2 +1HL2@@iDDL2 y{zz;;

errorLSB= Append@errorLSB,ErrLSBD;E;Export@"ErrorImRejLSB.dat", errorLSBD;

61

References

[1] H. H. Davè, A. Dubey, S. Thampi, R. P. Singh, “Submillimeter wave science and

applications”, in Proc. of the 28th URSI General Assembly, 2005, available online at:

http://www.ursi.org/Proceeding/ProcGA05/pdf/JB2.4(0610).pdf

[2] A. Barychev, “Superconductor-Insulator-Superconductor THz mixer integrated

with a superconducting flux-flow oscillator”, PhD thesis, Technische Universiteit

Delft, (2005).

[3] T. G. Phillips, “Prospects for high resolution spectroscopy from future space

platforms”, in Proc. of the Far-IR, Sub-MM and MM Detector Technology Workshop

(2002).

[4] See http://www.mpifr-bonn.mpg.de/staff/kmenten/APEX-flyer-KMM.pdf

[5] S. Payan et al., “A review of remote sensing techniques and related spectroscopy

problems”, in Competes Rendus Physique, 6, 825, (2005).

[6] See http://www.eso.org/projects/ALMA/specifications/FreqBands.html

[7] “Science with ALMA”, available online at:

http://www.eso.org/projects/ALMA/publications/papers/alma-science.pdf

[8] K. Kikuchi et al., “Simple FTS measurement system for submillimeter SIS

mixer”, Int. J. IR MM Waves, 23, 1019, (2002).

[9] D. Teyssier et al., “A multi-path far-infrared and sub-millimeter gas cell for

spectral tests of Herschel/HIFI”, in Proc. of the 15th ISSTT, 2004, session 7,

paper 25.

[10] A. R. Kerr et al., “Sideband calibration of millimetre-wave receivers”, ALMA

Memo 357, available online at: http://www.alma.nrao.edu/memos/html-

memos/alma357/memo357.pdf

[11] J. H. Hinken, “Superconductor electronics: fundamentals and microwave

applications”, Springer-Verlag (1989).

[12] J. Bardeen, L. N. Cooper, J. R. Schrieffer, “Theory of superconductivity”, Phys.

Rev. 108, 1175, (1957).

62

[13] S. Withington et al., “Multitone quantum simulations of saturating tunnel

junction mixers”, in J. Appl. Phys. 93, 9812, (2003).

[14] V.V. Vassilev, “Development of a sideband separating SIS mixer technology for

mm-wavelenghts”, PhD thesis, Chalmers University of Technology of

Göteborg, Sweden, (2003).

[15] F. P. Mena et al., “Construction of a side-band separating heterodyne mixer for

Band 9 of ALMA”, in Proc. of the 18th ISSTT, 2007.

[16] J. C. Albergotti, “Fourier Transform Spectroscopy Using a Michelson

Interferometer”, in American J. of Phys., 40, 1070, (1972).

[17] M. Bin et al., “A large throughput high resolution Fourier transform

spectrometer for submillimeter applications”, Int. J. IR MM Waves, 20, 383,

(1999).

[18] G. H. Tan, “Band 9 Cartridge technical specifications, FEND-40.02.09.00-002-

A-SPE”, ESO report, December 2003.

[19] J. R. Tucker and M. J. Feldman, “Quantum detection at millimeter

wavelenghts”, in Rev. of Modern Phys., 57, 1055, (1985).

[20] B. N. Taylor et al., “Guidelines for evaluating and expressing the uncertainty of

NIST measurement results”, NIST Technical Note 1297, (1994).

63

ACKNOWLEDGEMENTS Two years have gone and I am now at the end of my master…at last! There are

many people that made this possible, and for this I would like to thank them.

A thank you goes in particular to Prof. Reynier Peletier, who gave me the possibility

to attend this new master programme and who has followed my progress during

these two years. Thanks to all the “Kapteyner” that I met during this period, and to

Umut for keeping me company in our student room.

A big thanks goes to SRON, to Wolfgang Wild and to the whole ALMA group, in

which I was able to bring to completion the work presented in this thesis, and who

gave me the fantastic opportunity to attend this year’s ISSTT conference in

Pasadena. Thanks to the ALMA group, to the Kapteyn and to the G.U.F. for the

financial support.

I would like to thank in particular all the people that I worked in close contact with

during my final master project. Thanks to Andrey Baryshev and Ronald Hesper,

who proposed this master project and helped me a lot with its realization during the

last 8 months. Thank you Patricio for spending so many long days in the lab, for

your help and for answering all my questions. I wish to thank all the technicians of

the SRON workshop, and especially Klaas, for the way he taught me how to put a

design into practice, for the nice discussions we had, and for the soups at lunchtime.

A big hug to the “Little Italy” community of Groningen: to Sara, Fabrizio and

Giuseppe, to Filippo, Daniele, Alessio and Damiano; for all the moments spent

together, the nights when we watched movies and ate the renouned cold cuts from

Piacenza, for the fixed appointment on Saturdays on the Grote Markt to bite some

fish and chips... I’ll miss you!

An affectionate thanks to all my family, my dad and mom, Matteo and Martina, for

being always close to me and for backing all my choices. And a special thanks to

Roberto for always being there, for his help, for his encouragement, and for making

me happy.

Thank you all! Michela

64

RINGRAZIAMENTI

Due anni sono passati e ora sono alla fine del mio master…finalmente! Ci sono

molte persone che hanno reso possibile ciò e che per questo voglio ringraziare.

Un grazie particolare al prof. Reynier Peletier che mi ha dato l’opportunità di

frequentare questo nuovo master e che ha seguito il mio percorso durante questi

due anni. E grazie naturalmente a tutte le persone del Kapteyn che ho conosciuto.

Grazie alla SRON, a Wolfgang Wild e a tutto il gruppo di ALMA dove ho potuto

portare a termine il mio progetto finale di master e che mi hanno dato la possibilità

di prendere parte al ISSTT di quest’anno a Pasadena. Ringrazio il gruppo di ALMA,

il Kapteyn e il GUF per il supporto finanziario.

Vorrei ringraziare le persone con cui ho lavorato più a stretto contatto durante la

mia tesi. In particolare grazie ad Andrey Baryshev e Ronald Hesper per avermi

proposto questo progetto e avermi aiutato a portarlo a termine in questi ultimi 8

mesi. Grazie anche a Patricio per aver passato lunghe giornate in laboratorio con me

aiutandomi e rispondendo a ogni mia domanda. Grazie ai ragazzi del Workshop, e

in particolare a Klaas, per tutti gli insegnamenti su come si realizza un oggetto

partendo dal suo “disegno”, per le belle chiacchierate e le zuppe a pranzo.

Un abbraccio a tutta la “comunità italiana” conosciuta durante questi due anni a

Groningen. A Sara, Fabrizio e Giuseppe, a Filippo, Daniele, Alessio, Damiano: per

tutti i momenti passati assieme, le serate a guardare film mangiando i famosi salumi

piacentini, l’appuntamento del sabato al Grote Markt per gustare l’ormai leggendario

“Fish and Chips”…mi mancherete!

Grazie a tutta la mia famiglia, a papà, mamma, Matteo e Martina, per essermi stati

sempre vicino e per aver appoggiato ogni mia scelta.

E infine un grazie particolare a Roberto per esserci stato sempre, avermi aiutato,

consolato durante i momenti “no” e soprattutto reso felice.

Grazie a tutti!

Michela